Cramer's Rule

Given a set of linear equations

consider the determinant

Now multiply by , and use the property of determinants that multiplication by a constant is equivalent to multiplication of each entry in a single column by that constant, so

Another property of determinants enables us to add a constant times any column to any column and obtain the same determinant, so add times column 2 and times column 3 to column 1,

If , then (4) reduces to , so the system has nondegenerate solutions (i.e., solutions other than (0, 0, 0)) only if (in which case there is a family of solutions). If and , the system has no unique solution. If instead and , then solutions are given by

and similarly for

This procedure can be generalized to a set of equations so, given a system of linear equations

let

If , then nondegenerate solutions exist only if . If and , the system has no unique solution. Otherwise, compute

Then for . In the three-dimensional case, the vector analog of Cramer's rule is